Mathematics

# Evaluate the given integral.$\displaystyle \int { \cfrac { { x }^{ 2 } }{ 1+{ x }^{ 3 } } } dx$

##### SOLUTION
Let $t=1+{x}^{3}\Rightarrow\,dt=3{x}^{2}dx$

$\Rightarrow\,\dfrac{dt}{3}={x}^{2}dx$

$I=\displaystyle\int{\dfrac{{x}^{2}dx}{1+{x}^{3}}}$

$=\dfrac{1}{3}\displaystyle\int{\dfrac{dt}{t}}$

$=\dfrac{1}{3}\log{\left|t\right|}+c$    ........where $c$ is the constant of integration.

$=\dfrac{1}{3}\log{\left|1+{x}^{3}\right|}+c$    ............where $t=1+{x}^{3}$

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Subjective Medium Published on 17th 09, 2020
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